Subgroups — #27375426

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287690	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#20286382	C₂	1	2	yes	maximal, complemented
4	#20274307	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274181	C₂	1	2	yes	maximal, complemented
6	#20274109	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274075	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273910	C₂	1	2	yes	maximal, complemented
9	#20273820	C₂	1	2	yes	maximal, complemented
10	#20273601	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133249	—	2	2	no	—
13	#16811295	C₂²	1	2	yes	—
14	Q₈×D₂₁₂	C₂²	1	2	yes	—
15	#16811028	—	2	1	no	—
16	#16811028	C₂²	1	1	yes	characteristic
17	#16811000	—	2	1	no	—
18	#16811000	C₂²	1	1	yes	characteristic
19	#16810996	C₂²	1	2	yes	—
20	#16810970	C₂²	1	2	yes	—
21	#16810904	—	2	2	no	—
22	#16810904	C₂²	1	2	yes	—
23	#16810870	—	2	2	no	—
24	#16810833	C₂²	1	2	yes	—
25	#16810823	C₂²	1	2	yes	—
26	#16810819	—	2	2	no	—
27	#16810819	C₂²	1	2	yes	—
28	#16810788	—	2	2	no	—
29	#16810763	—	2	2	no	—
30	C₄×D₄₂₄	—	2	1	no	—
31	C₄×D₄₂₄	—	2	1	no	—
32	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
33	#16810702	C₂²	1	2	yes	—
34	#16810376	C₂²	1	1	yes	characteristic
35	#16810368	C₂²	1	2	yes	—
36	#16810366	C₂²	1	2	yes	—
37	#16810352	C₂²	1	2	yes	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#335	C₂²	1	1	yes	characteristic
42	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×#11719949	—	2	2	no	—
46	C₂×Q₈×D₅₃	—	2	2	no	—
47	#13133249	—	4	8	no	—
48	C₂×#11719925	—	2	2	no	—
49	C₂×#11719913	—	2	2	no	—
50	C₂×#11719913	—	2	2	no	—
51	C₂×#11719924	—	2	2	no	—
52	C₂×D₄₂₄	—	2	1	no	—
53	C₂×D₄₂₄	—	2	1	no	—
54	C₂×D₄₂₄	—	2	2	no	—
55	C₂×D₄₂₄	D₄	1	1	yes	characteristic
56	C₂×#11719912	—	2	2	no	—
57	#13133217	—	2	2	no	—
58	#13133217	C₂³	1	2	yes	—
59	#13133214	—	2	2	no	—
60	#13133214	C₂³	1	2	yes	—
61	#13133197	—	2	2	no	—
62	#13133197	—	4	2	no	—
63	#13133192	—	2	2	no	—
64	#13133192	—	4	2	no	—
65	C₄×D₂₁₂	—	2	2	no	—
66	C₄×D₂₁₂	—	2	2	no	—
67	C₄×D₂₁₂	C₂³	1	2	yes	—
68	#13133151	—	2	1	no	—
69	#13133151	—	2	1	no	—
70	#13133151	—	2	2	no	—
71	#13133151	D₄	1	1	yes	characteristic
72	C₄⋊#88322	C₂³	1	2	yes	—
73	#13133140	—	2	2	no	—
74	#13133139	—	2	2	no	—
75	#13133139	—	2	2	no	—
76	#13133139	—	4	2	no	—
77	#13133139	D₄	1	2	yes	—
78	#13133137	—	2	2	no	—
79	#13133129	—	2	2	no	—
80	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic
81	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
83	#107×C₅₃	D₄	1	1	yes	characteristic
84	C₄⋊C₈×C₅₃	—	2	1	no	—
85	C₄⋊C₈×C₅₃	—	2	1	no	—
86	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
88	#103×C₅₃	D₄	1	1	yes	characteristic
89	#103×C₅₃	D₄	1	1	yes	characteristic
90	C₄×C₈×C₅₃	—	2	1	no	—
91	C₄×C₈×C₅₃	—	2	1	no	—
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	#11719949	—	4	4	no	—
98	#11719949	—	8	2	no	—
99	Q₈×D₅₃	—	4	4	no	—
100	Q₈×D₅₃	—	8	2	no	—
101	C₂×D₂₁₂	—	2	2	no	—
102	C₂×D₂₁₂	—	2	2	no	—
103	C₂×D₂₁₂	—	2	2	no	—
104	C₂×D₂₁₂	—	4	2	no	—
105	C₂×D₂₁₂	—	4	2	no	—
106	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
107	C₂×C₄×D₅₃	—	2	2	no	—
108	C₂×C₄×D₅₃	—	4	2	no	—
109	C₂×C₄×D₅₃	—	4	2	no	—
110	C₂×Dic₁₀₆	—	2	2	no	—
111	C₂×Dic₁₀₆	—	4	2	no	—
112	Q₁₆×C₅₃	D₈	1	4	yes	complemented
113	C₂×C₈×C₅₃	—	2	1	no	—
114	C₂×C₈×C₅₃	—	2	1	no	—
115	C₂×C₈×C₅₃	—	2	1	no	—
116	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	—	2	1	no	—
122	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₅₃	—	2	1	no	—
125	C₄²×C₅₃	—	2	1	no	—
126	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
127	#11719925	—	4	4	no	—
128	#11719924	—	4	4	no	—
129	#11719921	—	2	2	no	—
130	#11719921	—	4	2	no	—
131	#11719921	—	4	2	no	—
132	C₄⋊Dic₅₃	—	2	2	no	—
133	C₄⋊Dic₅₃	—	2	2	no	—
134	C₄⋊Dic₅₃	—	4	2	no	—
135	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
136	C₂×#88322	—	2	2	no	—
137	D₄₂₄	—	2	2	no	—
138	D₄₂₄	—	4	2	no	—
139	D₄₂₄	—	8	1	no	—
140	D₄₂₄	—	8	2	no	—
141	#11719913	—	4	4	no	—
142	#11719913	—	8	2	no	—
143	#11719912	—	4	4	no	—
144	C₂²×D₅₃	—	2	2	no	—
145	C₂²×D₅₃	—	4	2	no	—
146	Q₈×C₅₃	—	2	1	no	—
147	Q₈×C₅₃	—	2	1	no	—
148	Q₈×C₅₃	C₂×D₈	1	2	yes	—
149	Q₈×C₅₃	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₅₃	—	2	1	no	—
151	C₂×C₄×C₅₃	—	2	1	no	—
152	C₂×C₄×C₅₃	—	2	1	no	—
153	C₂×C₄×C₅₃	—	2	1	no	—
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×Dic₅₃	—	2	2	no	—
158	C₂×Dic₅₃	—	4	2	no	—
159	D₂₁₂	—	2	2	no	—
160	D₂₁₂	—	2	2	no	—
161	D₂₁₂	—	4	2	no	—
162	D₂₁₂	—	4	2	no	—
163	D₂₁₂	—	4	2	no	—
164	D₂₁₂	—	4	2	no	—
165	D₂₁₂	—	8	4	no	—
166	D₂₁₂	—	8	2	no	—
167	D₂₁₂	—	16	2	no	—
168	C₄×D₅₃	—	4	2	no	—
169	C₄×D₅₃	—	4	2	no	—
170	C₄×D₅₃	—	8	4	no	—
171	C₄×D₅₃	—	8	4	no	—
172	Dic₁₀₆	—	4	2	no	—
173	Dic₁₀₆	—	4	2	no	—
174	Dic₁₀₆	—	8	4	no	—
175	C₈×C₅₃	—	2	1	no	—
176	C₈×C₅₃	—	2	1	no	—
177	C₈×C₅₃	—	4	1	no	—
178	C₈×C₅₃	—	4	1	no	—
179	C₈×C₅₃	—	4	1	no	—
180	C₈×C₅₃	C₂×D₈	1	2	yes	—
181	C₈×C₅₃	#137	1	2	yes	complemented
182	#88322	—	4	2	no	—
183	#19889	—	53	1	no	maximal, Sylow
184	C₂²×C₅₃	D₄²	1	1	yes	characteristic
185	C₂×D₅₃	—	4	2	no	—
186	C₂×D₅₃	—	4	2	no	—
187	C₂×D₅₃	—	4	2	no	—
188	C₂×D₅₃	—	8	2	no	—
189	C₂×D₅₃	—	8	2	no	—
190	C₂×D₅₃	—	8	2	no	—
191	C₄×C₅₃	—	2	2	no	—
192	C₄×C₅₃	—	2	2	no	—
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	—	4	1	no	—
195	C₄×C₅₃	—	4	1	no	—
196	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
197	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
198	C₄×C₅₃	#576	1	1	yes	characteristic
199	C₄×C₅₃	#577	1	1	yes	characteristic
200	Dic₅₃	—	4	2	no	—
201	Dic₅₃	—	8	2	no	—
202	#3391	—	53	1	no	—
203	Q₈×D₈	—	53	1	no	—
204	#3278	—	53	2	no	—
205	#1706	—	53	1	no	—
206	#1685	—	53	2	no	—
207	#1674	—	53	1	no	—
208	#1668	—	53	1	no	—
209	#1641	—	53	2	no	—
210	#1627	—	53	2	no	—
211	#1589	—	53	1	no	—
212	C₈×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
213	C₂×C₅₃	#3278	1	1	yes	characteristic
214	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
215	C₂×C₅₃	#3290	1	1	yes	characteristic
216	D₅₃	—	8	2	no	—
217	D₅₃	—	16	2	no	—
218	C₂×#137	—	106	2	no	—
219	#550	—	53	2	no	—
220	D₄×Q₈	—	53	2	no	—
221	#500	—	53	1	no	—
222	#500	—	106	1	no	—
223	#493	—	53	1	no	—
224	#493	—	106	1	no	—
225	#492	—	53	2	no	—
226	#486	—	53	2	no	—
227	#474	—	53	2	no	—
228	#474	—	106	2	no	—
229	#468	—	106	2	no	—
230	#462	—	53	2	no	—
231	#460	—	53	2	no	—
232	#459	—	53	2	no	—
233	#459	—	106	2	no	—
234	#452	—	106	2	no	—
235	#448	—	106	2	no	—
236	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
237	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
238	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
239	C₄×D₈	—	53	1	no	—
240	C₄×D₈	—	106	1	no	—
241	C₄×D₈	—	106	1	no	—
242	#436	—	53	2	no	—
243	#335	—	53	1	no	—
244	#335	C₂×D₅₃	1	1	yes	characteristic
245	#332	—	53	2	no	—
246	#331	—	53	2	no	—
247	Q₈⋊C₈	C₂×D₅₃	1	1	yes	characteristic
248	Q₈⋊C₈	C₂×D₅₃	1	1	yes	characteristic
249	D₄⋊C₈	—	53	2	no	—
250	C₈²	C₂×D₅₃	1	1	yes	characteristic
251	C₅₃	#19889	1	1	yes	characteristic, complemented, Sylow
252	C₂×#41	—	106	2	no	—
253	C₂²×Q₈	—	106	2	no	—
254	#137	—	212	8	no	—
255	C₂×Q₁₆	—	106	2	no	—
256	C₂×Q₁₆	D₂₁₂	1	1	yes	characteristic
257	C₂×SD₁₆	—	106	2	no	—
258	C₂×SD₁₆	—	106	2	no	—
259	C₂×SD₁₆	—	106	2	no	—
260	C₂×D₈	—	53	1	no	—
261	C₂×D₈	—	106	2	no	—
262	C₂×D₈	—	106	1	no	—
263	C₂×D₈	—	106	1	no	—
264	C₂×M₁₆	—	106	2	no	—
265	C₄⋊Q₈	—	53	2	no	—
266	C₄⋊Q₈	—	106	2	no	—
267	#127	—	53	2	no	—
268	#127	—	106	2	no	—
269	C₂²⋊Q₈	—	106	2	no	—
270	C₂²⋊Q₈	—	212	2	no	—
271	#121	—	106	2	no	—
272	#121	—	212	2	no	—
273	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
274	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
275	C₄×D₄	—	53	2	no	—
276	C₄×D₄	—	106	2	no	—
277	C₄×D₄	—	106	2	no	—
278	#107	—	53	1	no	—
279	#107	—	106	2	no	—
280	#107	—	106	1	no	—
281	#107	—	106	1	no	—
282	#107	D₂₁₂	1	1	yes	characteristic
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	—	53	2	no	—
286	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
287	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
288	#103	—	106	2	no	—
289	#103	D₂₁₂	1	1	yes	characteristic
290	#103	D₂₁₂	1	1	yes	characteristic
291	#102	—	53	2	no	—
292	#102	—	106	2	no	—
293	#102	—	106	2	no	—
294	#102	—	106	2	no	—
295	#102	—	212	2	no	—
296	C₂²⋊C₈	—	106	2	no	—
297	C₄×C₈	—	2	1	no	—
298	C₄×C₈	—	2	1	no	—
299	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
300	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
301	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
302	#41	—	424	2	no	—
303	#41	—	212	4	no	—
304	C₂×Q₈	—	106	2	no	—
305	C₂×Q₈	—	424	2	no	—
306	C₂×Q₈	—	212	2	no	—
307	C₂×Q₈	—	212	4	no	—
308	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
309	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
310	C₂×D₄	—	53	2	no	—
311	C₂×D₄	—	106	2	no	—
312	C₂×D₄	—	106	2	no	—
313	C₂×D₄	—	106	2	no	—
314	C₂×D₄	—	212	2	no	—
315	C₂×D₄	—	212	2	no	—
316	C₂²×C₄	—	106	2	no	—
317	C₂²×C₄	—	212	2	no	—
318	C₂²×C₄	—	212	2	no	—
319	Q₁₆	—	212	4	no	—
320	Q₁₆	D₄₂₄	1	4	yes	complemented
321	SD₁₆	—	424	2	no	—
322	SD₁₆	—	212	4	no	—
323	SD₁₆	—	212	4	no	—
324	D₈	—	106	2	no	—
325	D₈	—	424	1	no	—
326	D₈	—	424	2	no	—
327	D₈	—	212	2	no	—
328	M₁₆	—	212	4	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	106	2	no	—
333	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
334	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
335	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
336	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	53	2	no	—
340	C₄⋊C₄	—	106	2	no	—
341	C₄⋊C₄	—	106	2	no	—
342	C₄⋊C₄	—	212	2	no	—
343	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
344	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
345	C₂²⋊C₄	—	106	2	no	—
346	C₂²⋊C₄	—	212	2	no	—
347	C₂²⋊C₄	—	212	2	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	2	1	no	—
350	C₄²	C₂³×D₅₃	1	1	yes	characteristic
351	C₂³	—	106	2	no	—
352	C₂³	—	212	2	no	—
353	Q₈	—	2	1	no	—
354	Q₈	—	2	1	no	—
355	Q₈	—	424	4	no	—
356	Q₈	—	212	2	no	—
357	Q₈	—	212	2	no	—
358	Q₈	C₂×D₄₂₄	1	2	yes	—
359	Q₈	C₂×D₄₂₄	1	2	yes	—
360	D₄	—	848	2	no	—
361	D₄	—	106	2	no	—
362	D₄	—	106	2	no	—
363	D₄	—	424	2	no	—
364	D₄	—	424	4	no	—
365	D₄	—	212	2	no	—
366	D₄	—	212	2	no	—
367	D₄	—	212	2	no	—
368	D₄	—	212	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	106	2	no	—
374	C₂×C₄	—	424	4	no	—
375	C₂×C₄	—	424	4	no	—
376	C₂×C₄	—	212	2	no	—
377	C₂×C₄	—	212	2	no	—
378	C₂×C₄	—	212	2	no	—
379	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
381	C₂×C₄	#13133263	1	1	yes	characteristic
382	C₈	—	2	1	no	—
383	C₈	—	2	1	no	—
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	212	2	no	—
388	C₈	C₂×D₄₂₄	1	2	yes	—
389	C₈	#13133249	1	2	yes	complemented
390	C₂²	—	424	2	no	—
391	C₂²	—	424	2	no	—
392	C₂²	—	424	2	no	—
393	C₂²	—	212	2	no	—
394	C₂²	—	212	2	no	—
395	C₂²	—	212	2	no	—
396	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	2	2	no	—
399	C₄	—	2	2	no	—
400	C₄	—	4	1	no	—
401	C₄	—	4	1	no	—
402	C₄	—	424	2	no	—
403	C₄	—	212	2	no	—
404	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
405	C₄	C₂×#13133249	1	1	yes	characteristic
406	C₄	#16811428	1	1	yes	characteristic
407	C₄	#16811430	1	1	yes	characteristic
408	C₂	—	848	2	no	—
409	C₂	—	424	2	no	—
410	C₂	#20286382	1	1	yes	characteristic
411	C₂	D₄×D₄₂₄	1	1	yes	characteristic
412	C₂	#20286502	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	D₄²
prime radical	#20287690	C₂
hypercenter	C₈×Q₁₆	D₅₃
hypocenter	C₅₃	#19889
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. Q₈×D₄₂₄	C₂	semidirect
3. #20286382	C₂	semidirect
4. #20274307	C₂	semidirect
5. #20274181	C₂	semidirect
6. #20274109	C₂	semidirect
7. #20274075	C₂	semidirect
8. #20273910	C₂	semidirect
9. #20273820	C₂	semidirect
10. #20273601	C₂	semidirect
11. C₈×Q₁₆×C₅₃	C₂	semidirect
112. Q₁₆×C₅₃	D₈	semidirect
181. C₈×C₅₃	#137	semidirect
212. C₈×Q₁₆	D₅₃	semidirect
251. C₅₃	#19889	semidirect
320. Q₁₆	D₄₂₄	semidirect
389. C₈	#13133249	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20287690	2
Q₈×D₄₂₄	2
#20286382	2
#20274307	2
#20274181	2
#20274109	2
#20274075	2
#20273910	2
#20273820	2
#20273601	2
C₈×Q₁₆×C₅₃	2
#19889	53